Could quicker Rumsfeld exit have kept GOP in power?

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WASHINGTON (CNN) -- Still smarting from the rebuke they suffered in last week's elections, Republicans were split Sunday over whether ousting Defense Secretary Donald Rumsfeld earlier might have kept their party in power.

"If Rumsfeld had been out, you bet it would have made a difference," Sen. Arlen Specter, R-Pennsylvania, told CNN. "I would still be chairman of the Judiciary Committee."

Actually, he is. Specter remains Judiciary Committee chairman during the lame-duck session of Congress until the Democrats take over in January.

"He should have done it as soon as he made up his mind, and that's a hard thing to calculate, but it's highly doubtful that he made up his mind between the time the election returns came in on Tuesday, and Wednesday, when Rumsfeld was out," Specter said on CNN's "Late Edition with Wolf Blitzer."

Former House Speaker Newt Gingrich was adamant that the move's timing was a mistake.

"If the president had replaced Rumsfeld two weeks ago, the Republicans would still control the Senate and they would probably have 10 more House members," Gingrich said.

But White House operatives appeared less persuaded that their boss had made a tactical error.

"I could argue the politics of it either way," White House Chief of Staff Joshua Bolten told CNN. "It might have been disheartening for people to see a sudden change at that moment, two or three weeks before the election. But however you view the politics of it, it doesn't matter. The president wasn't ready to make the decision until the last week. And he was determined not to inject politics into one of the most important national security decisions he has got to make."

Bolten added, "If it affected votes one way or the other -- maybe it did. I'm skeptical about what the effect would have been. But I think the president did the right thing in the right timing."

Counselor to the President Dan Bartlett said the election results did not necessarily reflect voters' views on Rumsfeld's stewardship of the war in Iraq.

"I don't necessarily buy the calculation that he was the difference in the election," Bartlett told "Fox News Sunday."

Had Bush made the move sooner, "it would have looked desperate, " Bartlett added. "I think that would have weakened the president and Republican support going down the stretch of this campaign."